Question

Use the properties of exponents to generate an equivalent expression for 3√8 divided by √2
Find the cube root of 8. Then, rewrite rational exponents as radicals and simplify.

Answer 1

To find an equivalent expression for \(\frac{\sqrt[3]{8}}{\sqrt{2}}\), we will proceed through the following steps.

1. **Find the cube root of 8:**
\[
\sqrt[3]{8} = 2
\]
This is because \(2^3 = 8\).

2. **Rewrite the expression:**
The expression \(\frac{\sqrt[3]{8}}{\sqrt{2}}\) can now be simplified as:
\[
\frac{2}{\sqrt{2}}
\]

3. **Rewrite \(\sqrt{2}\) as a rational exponent:**
Recall that \(\sqrt{2}\) can be expressed as:
\[
\sqrt{2} = 2^{1/2}
\]
Therefore, the expression becomes:
\[
\frac{2}{2^{1/2}}
\]

4. **Use the properties of exponents to simplify:**
We can rewrite \(2\) as \(2^{1}\) so that:
\[
\frac{2^{1}}{2^{1/2}} = 2^{1 - 1/2} = 2^{1/2}
\]

5. **Rewrite \(2^{1/2}\) as a radical:**
Thus:
\[
2^{1/2} = \sqrt{2}
\]

So the final equivalent expression for \(\frac{\sqrt[3]{8}}{\sqrt{2}}\) is:
\[
\sqrt{2}
\]